The Race Tournaments are unlocked after you complete Driving School, and can be found in each of the major cities. There's nothing exceptionally unique about these races; when you find the location where they're given, step into the circle to get a menu of available races, then select one and get going. In order to get a 100% game completion percentage, you'll need to get first place in all of the races in all four of the race tournaments. Obviously enough, this is easier to do when you have a very high Driving/Bike/Flying skill, so you may want to save these races until later in the game.
The races can be fairly difficult, if only because your competitors will often run near-perfect races. You do have a bit of a speed advantage, though, so if you can knock them around or get them off the track, you'll usually be able to build up a lead. The most basic advice we can give is precisely that: build up a lead, preferably early in the race, and then don't make mistakes. Some of your ability to do this will rely on your skill with whatever type of vehicle you're using in the race, but part of it will rely on your patience for retrying a race over and over, as no amount of pure description can prepare you for the twists and turns of each individual race, of which there are over 20 in all. Practice, then, and repeat each race until you come in first.
In most of the races, your worst nightmare will be oncoming and outgoing traffic; inadvertant collisions will be the source of many a spill or flip and a restarted race. Try to go off-road whenever it's feasible to do so without actually increasing your risk; even staying on the shoulder or sidewalk is sometimes preferably to being on the street. If you're confident of your ability to thread the needle, you can also ride down the stripes in the middle of the street, but tapping into one of the oncoming cars will usually sap all of your speed, so this is a risky proposition.
In general, though, you can afford to slow down when you need to, especially if it's in the service of avoiding a collision or something that would ruin your entire run. Occasionally, though, you will take a superspill that will push you to the back of the pack and effectively make it impossible to come in first. If this occurs, you can quickly get back to the race-selection menu by getting out of your car and blowing it up. (In most races, blowing up your opponents will also disqualify you, although this is a bit harder to do.)
You can find the Los Santos races in Little Mexico. The most difficult race here is Freeway, when you'll have to complete a circuit on the eponymous roadway with a Super GT. The Super GT is a powerful car, and fast, but a fragile one, and will explode with the barest provocation. There's nothing quite as deflating as getting all the way through a race and then blowing up just before the finish line, so try not to knock yourself around too much.
The San Fierro races are located near the Wang Cars dealership, near your garage save point. Many of these races will actually take place in the backwoods areas south of San Fierro, rather than in the city itself, including the devestatingly difficult Country Endurance race. Although you have a fast car for this latter race - a Bullet - you have to go almost seven miles in it, and the Bullet is one of the physically weakest cars in the entire game, meaning that you'll have to be ponderously careful not to hit any obstacles or cars.
There are two race tournaments in Las Venturas. The first is the Freight Depot race tournament, which possesses only four ground races, two of which are motorcycle challenges. None of these are as difficult as the most challenging races in the other cities, so hit them a few times until you get first place.
The second tournament will give you some headaches, however, as they're all flying races. You can find these races just to the right of the main pilot's entrance to the airport. These actually aren't races at all, but rather timed missions that require you to pass through a series of coronas (much like in the flight school missions). Unfortunately, you aren't given a timer or a target time for these, so you'll just have to do the best you can, given the sometimes over-sensitive flight controls in the game. Don't forget to use your L2 and R2 buttons to turn as often as possible; if you rely exclusively on the analog stick, you'll often find yourself oversteering.